# Can quantum states have negative probability?

I know that when a qubit is described in a superposition state as $$\alpha|0⟩ + \beta|1⟩$$, then the probability of measuring $$|0⟩$$ is $$\alpha^2$$ and probability of measuring $$|1⟩$$ is $$\beta^2$$ and $$\alpha$$ and $$\beta$$ belong to the complex number set, $$C$$.

So let's say that, the amplitude of state, say, $$|0⟩$$ is $$\alpha=i$$, which means that the probabilty of measuring the $$|0⟩$$ state will be $$\alpha^2=i^2=-1$$.

So, my question is, does the mean that the probability here can also be negative? Since, as far as I knew, negative probability does not exist? Or is my understanding wrong?

– J.G.
Jul 30, 2020 at 15:35

Just to add, I sometime come accros notion of negative probability in quantum mechanics. However, this misunderstanding. As you mentioned, each qubit can be writen as $$|q\rangle = \alpha|0\rangle + \beta|1\rangle,$$ where $$\alpha, \beta \in \mathbb{C}$$.

So, we can have for example a qubit $$|q\rangle = \frac{1}{\sqrt{2}}(|0\rangle - \beta|1\rangle).$$

It may seem that probability of measuring state $$|1\rangle$$ is negative, but the coefficient $$-\frac{1}{\sqrt{2}}$$ is a complex amplitude, not the probability itself.

The probabilities are, as already mentioned by Norbert Schuch, calculated as $$P(|0\rangle) = |\alpha|^2$$ $$P(|1\rangle) = |\beta|^2$$ Hence they are non-negative.

The negative sign in my example is caused by non-zero phase of the qubit which is $$\pi$$ in this case (note that $$\mathrm{e}^{i \pi} = -1$$).

The probability is $$|\alpha|^2$$, not $$\alpha^2$$, and thus always non-negative.

Though we don't come across negative probabilities in a quantum computation problem in the general sense, there is a historic context on the discussion and debate around negative probabilities in quantum mechanics.

In 1942, Paul Dirac wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative energies and negative probabilities. The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman introduced ghosts as "negative probability" in perturbative gauge theories. The main purpose of the ghosts is to cancel the contributions from unphysical polarisations of gauge fields in loops.

Another example is known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities. For this reason, it has later been better known as the Wigner quasiprobability distribution. The Wigner distribution function is routinely used in physics nowadays and provides the cornerstone of phase-space quantization. Its negative features are an asset to the formalism and often indicate quantum interference.

However, one never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables.

There are two works of Feynman about negative probabilities.

R. P. Feynman, Negative probability in Quantum implications: Essays in Honor of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987), Chap. 13, pp 235 – 248.

R. P. Feynman, Simulating physics with computers (Chapter 6), Int. J. Theor. Phys., 21, 467 – 488 (1982).